Armature-control
Consider the armature-controlled d.c. motor shown in Fig.
In this system,
Rₐ = resistance of armature (ohm)
Lₐ = inductance of armature winding (H).
iₐ = armature current (A).
if = field current (A).
eₐ = applied armature voltage (V).
eb = back emf (volts).
Tₘ = torque developed by motor (Nm).
θ = angular displacement of motor-shaft (rad).
J = equivalent moment of inertia of motor and load referred to motor shaft (kg-m^2).
f₀ = equivalent viscous friction coefficient of motor and load referred to motor shaft(Nm/(rad/s))

In servo applications, the d.c. motors are generally used in the linear range of the
magnetization curve. Therefore, the air gap flux φ is proportional of the field current
i.e.,  φ= Kf*if    where, Kf is a constant.


The torque Tₘ developed by the motor is proportional to the product of the armature
current and air gap flux, 
i.e., Tₘ=K₁*Kf*if*iₐ   where, K₁ is a constant.

In the armature-controlled d.c. motor, the field current is kept constant, so that eqn
can be written as

Тₘ= Kₜ*iₐ

where Kₜ, is known as the motor torque constant.
The motor back emf being proportional to speed is given as

eb = Kb dθ/dt
where Kb is the back emf constant.

The differential equation of the armature circuit is


The torque equation is

(2.51)
dt
Taking the Laplace transforms of eqns (2.48) to 12.50), assuming zero initial conditions,
we get
2.,52)
...(253)
(Ja + fe) 6Cs) = T)=K.I()
(2.54)
From ens. (2.51) to (2.53), the transfer function of the system is obtained as

G
OL)
E(s oR,+sL XJs fo+K,K,
...(2.55)
The block diagram representation of equation. (2.53) is shown in Fig. 2.22 (a) where the
circular block representing the differencing action is known as the summing point. Equation

(2.54) is represented by a block shown in Fig. 2 22 (b).

Summing
point

Tako o
point

Fig. 2.22. Block diagram of armature-controlled d.c. motor
Figure 2.22 (c) represents equation. (2.52) where a signal is taken off from a fake-off point
and fed to the feedback block K . Pig 2.22 (d) is the complete block diagram of the system
under consideration, obtained by connecting the block diagram shown in Fig. 2.22 (a), (b) and
(o). It may be pointed out here that when a signal is taken from the output of a bl this does
not affect the output as per assumption 1 of the procedure for driving transfer functions advanced
earlier
However, it should be noted that the block diagram of the system 5under consideration
can be directly obtained from the physical system of Fig. 2:21 by using the transfer functions of
simple electrical and mechanical networks derived already. The voltage applied to the armature
circuit is B ) which is opposed by the back emf (EC)). The net voltage - acts on a
linear circuit comprised of resistance and inductance in series, having the transfer function
1L+R The result is an armature current 1.).